Expected Value

“Expected value” – or EV – is the amount a player can expect to win or lose if they were to place a bet on the same odds many times over. For example, if you were to bet $10 on heads in a coin toss, and you were to receive $11 every time you got it right, the EV would be 0.5. This means that if you were to make the same bet on heads over and over again, you can expect to win an average of $0.50 for each bet of $10.

The expected value of a discrete random variable is the probability-weighted average of all possible values. To calculate this, each possible value is multiplied by its probability of occurring. The resulting number is the expected value. For continuous random variables, you do the same calculation but replace the sum with an integral and the probabilities with probability densities.

It is important to note that expected value calculations doesn’t work well for the sports bettor when the random variables have distributions with “heavy tails”. In probability theory, heavy-tailed distributions are probability distributions where the tails are not exponentially bounded, i.e. the tails are heavier than the exponential distribution. You should be on the lookout for both fat-tailed distributions, long-tailed distributions and subexponential distributions, since they can all mess with your results and make you lose money on sports betting if you allow them to fool you while making expected value calculations.

How to calculate expected value for your bet

Most sports bettors that calculate expected value will use this simple formula:

(probability of wining) x (amount won per bet) – (probability of losing) x (amount lost per bet)

Use the decimal odds for each outcome.

The potential winnings for each outcome is calculated by multiplying your wager by the decimal, and then subtract the wagered amount.

To calculate the probability of an outcome, divide 1 by the odds.

Example

Team A (1.263) plays Team B (13.500). Draw is at 6.500.

A €10 bet on a Team B win would give you €125 if you won. The probability of you winning is 0.074.

The probability of you not winning is the likelihood of a Team A + the likelihood of a draw, which is 0.792 + 0.154 = 0.946.

With these numbers, your calculation of expected value will look like this:

(0.074 x €125) – (0.946 x €10) = – €0.20

As you can see, the expected value for this is negative. According to the expected value, you can expect to, on average, lose €0.20 for each €10 you bet.

Using expected value to find suitable opportunities

Many punters will use the expected value to find opportunities where they disagree with the bookmaker. Let’s for instance say that bookmaker’s odds imply that Team YYY has a 10 percent of winning. You, on the other hand, have analyzed the situation – perhaps with the aid of Poisson distribution or a similar method – and come to the conclusion that Team YYY actually have a 15 percent chance of winning. Now, the expected value for betting on a Team YYY win becomes really nice and high.